Internal problem ID [9506]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1931.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} a x^{\prime }\left (t \right )&=\left (b -c \right ) y \left (t \right ) z \left (t \right )\\ b y^{\prime }\left (t \right )&=\left (-a +c \right ) z \left (t \right ) x \left (t \right )\\ c z^{\prime }\left (t \right )&=\left (a -b \right ) x \left (t \right ) y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 2.093 (sec). Leaf size: 1356
dsolve([a*diff(x(t),t)=(b-c)*y(t)*z(t),b*diff(y(t),t)=(c-a)*z(t)*x(t),c*diff(z(t),t)=(a-b)*x(t)*y(t)],[x(t), y(t), z(t)], singsol=all)
\begin{align*} \{z \left (t \right ) = 0\} \\ \{y \left (t \right ) = 0\} \\ \{x \left (t \right ) = c_{1}\} \\ \end{align*} \begin{align*} \{z \left (t \right ) = 0\} \\ \{y \left (t \right ) = c_{1}\} \\ \{x \left (t \right ) = 0\} \\ \end{align*} \begin{align*} \{z \left (t \right ) = c_{1}\} \\ \{y \left (t \right ) = 0\} \\ \{x \left (t \right ) = 0\} \\ \end{align*} \begin{align*} \left \{z \left (t \right ) = \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}-\frac {2 b a \left (b a -a c -b c +c^{2}\right )}{\sqrt {b a \left (b a -a c -b c +c^{2}\right ) \left (-4 \textit {\_a}^{4} a^{2} b^{2}+8 \textit {\_a}^{4} a^{2} b c -4 \textit {\_a}^{4} a^{2} c^{2}+8 \textit {\_a}^{4} a \,b^{2} c -16 \textit {\_a}^{4} a b \,c^{2}+8 \textit {\_a}^{4} a \,c^{3}-4 \textit {\_a}^{4} b^{2} c^{2}+8 \textit {\_a}^{4} b \,c^{3}-4 \textit {\_a}^{4} c^{4}+16 c_{2} \textit {\_a}^{2} a^{2} b^{2}-32 c_{2} \textit {\_a}^{2} a^{2} b c +16 c_{2} \textit {\_a}^{2} a^{2} c^{2}-32 c_{2} \textit {\_a}^{2} a \,b^{2} c +64 c_{2} \textit {\_a}^{2} a b \,c^{2}-32 c_{2} \textit {\_a}^{2} a \,c^{3}+16 c_{2} \textit {\_a}^{2} b^{2} c^{2}-32 c_{2} \textit {\_a}^{2} b \,c^{3}+16 c_{2} \textit {\_a}^{2} c^{4}-16 c_{2}^{2} a^{2} b^{2}+32 c_{2}^{2} a^{2} b c -16 c_{2}^{2} a^{2} c^{2}+32 c_{2}^{2} a \,b^{2} c -64 c_{2}^{2} a b \,c^{2}+32 c_{2}^{2} a \,c^{3}-16 c_{2}^{2} b^{2} c^{2}+32 c_{2}^{2} b \,c^{3}-16 c_{2}^{2} c^{4}+b a c_{1} \right )}}d \textit {\_a} \right )+t +c_{3} \right ), z \left (t \right ) = \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {2 b a \left (b a -a c -b c +c^{2}\right )}{\sqrt {b a \left (b a -a c -b c +c^{2}\right ) \left (-4 \textit {\_a}^{4} a^{2} b^{2}+8 \textit {\_a}^{4} a^{2} b c -4 \textit {\_a}^{4} a^{2} c^{2}+8 \textit {\_a}^{4} a \,b^{2} c -16 \textit {\_a}^{4} a b \,c^{2}+8 \textit {\_a}^{4} a \,c^{3}-4 \textit {\_a}^{4} b^{2} c^{2}+8 \textit {\_a}^{4} b \,c^{3}-4 \textit {\_a}^{4} c^{4}+16 c_{2} \textit {\_a}^{2} a^{2} b^{2}-32 c_{2} \textit {\_a}^{2} a^{2} b c +16 c_{2} \textit {\_a}^{2} a^{2} c^{2}-32 c_{2} \textit {\_a}^{2} a \,b^{2} c +64 c_{2} \textit {\_a}^{2} a b \,c^{2}-32 c_{2} \textit {\_a}^{2} a \,c^{3}+16 c_{2} \textit {\_a}^{2} b^{2} c^{2}-32 c_{2} \textit {\_a}^{2} b \,c^{3}+16 c_{2} \textit {\_a}^{2} c^{4}-16 c_{2}^{2} a^{2} b^{2}+32 c_{2}^{2} a^{2} b c -16 c_{2}^{2} a^{2} c^{2}+32 c_{2}^{2} a \,b^{2} c -64 c_{2}^{2} a b \,c^{2}+32 c_{2}^{2} a \,c^{3}-16 c_{2}^{2} b^{2} c^{2}+32 c_{2}^{2} b \,c^{3}-16 c_{2}^{2} c^{4}+b a c_{1} \right )}}d \textit {\_a} \right )+t +c_{3} \right )\right \} \\ \left \{y \left (t \right ) = -\frac {\sqrt {2}\, \sqrt {z \left (t \right ) b \left (b a -a c -b^{2}+b c \right ) \left (\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) a b -\sqrt {4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a^{2} b^{2}-4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a^{2} b c -4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a \,b^{2} c +4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a b \,c^{2}+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} a^{2} b^{2}}\right ) c}}{2 z \left (t \right ) b \left (b a -a c -b^{2}+b c \right )}, y \left (t \right ) = \frac {\sqrt {2}\, \sqrt {z \left (t \right ) b \left (b a -a c -b^{2}+b c \right ) \left (\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) a b -\sqrt {4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a^{2} b^{2}-4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a^{2} b c -4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a \,b^{2} c +4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a b \,c^{2}+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} a^{2} b^{2}}\right ) c}}{2 z \left (t \right ) b \left (b a -a c -b^{2}+b c \right )}, y \left (t \right ) = -\frac {\sqrt {2}\, \sqrt {z \left (t \right ) b \left (b a -a c -b^{2}+b c \right ) \left (\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) a b +\sqrt {4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a^{2} b^{2}-4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a^{2} b c -4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a \,b^{2} c +4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a b \,c^{2}+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} a^{2} b^{2}}\right ) c}}{2 z \left (t \right ) b \left (b a -a c -b^{2}+b c \right )}, y \left (t \right ) = \frac {\sqrt {2}\, \sqrt {z \left (t \right ) b \left (b a -a c -b^{2}+b c \right ) \left (\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) a b +\sqrt {4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a^{2} b^{2}-4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a^{2} b c -4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a \,b^{2} c +4 \left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )^{2} a b \,c^{2}+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} a^{2} b^{2}}\right ) c}}{2 z \left (t \right ) b \left (b a -a c -b^{2}+b c \right )}\right \} \\ \left \{x \left (t \right ) = \frac {c \left (\frac {d}{d t}z \left (t \right )\right )}{a y \left (t \right )-b y \left (t \right )}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.553 (sec). Leaf size: 1285
DSolve[{a*x'[t]==(b-c)*y[t]*z[t],b*y'[t]==(c-a)*z[t]*x[t],c*z'[t]==(a-b)*x[t]*y[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {\sqrt {2} \sqrt {b c_1 (b-c)} \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right )}{\sqrt {a (a-c)}} \\ y(t)\to -\frac {\sqrt {2} \sqrt {-b c_1 (b-c) \left (-1+\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to \frac {\sqrt {2} \sqrt {c c_2 (b-c) \text {dn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2}}{\sqrt {c} \sqrt {b-c}} \\ x(t)\to \frac {\sqrt {2} \sqrt {b c_1 (b-c)} \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right )}{\sqrt {a (a-c)}} \\ y(t)\to \frac {\sqrt {2} \sqrt {-b c_1 (b-c) \left (-1+\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to -\frac {\sqrt {2} \sqrt {c c_2 (b-c) \text {dn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2}}{\sqrt {c} \sqrt {b-c}} \\ x(t)\to \frac {\sqrt {2} \sqrt {b c_1 (b-c)} \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right )}{\sqrt {a (a-c)}} \\ y(t)\to -\frac {\sqrt {2} \sqrt {-b c_1 (b-c) \left (-1+\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to -\frac {\sqrt {2} \sqrt {c c_2 (b-c) \text {dn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2}}{\sqrt {c} \sqrt {b-c}} \\ x(t)\to \frac {\sqrt {2} \sqrt {b c_1 (b-c)} \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right )}{\sqrt {a (a-c)}} \\ y(t)\to \frac {\sqrt {2} \sqrt {-b c_1 (b-c) \left (-1+\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to \frac {\sqrt {2} \sqrt {c c_2 (b-c) \text {dn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2}}{\sqrt {c} \sqrt {b-c}} \\ \end{align*}